Summary of equations chapters 7. To make current flow you have to push on the charges. For most materials:
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1 Summry of equtions chpters 7. To mke current flow you hve to push on the chrges. For most mterils: J E E [] The resistivity is prmeter tht vries more thn 4 orders of mgnitude between silver (.6E-8 Ohm.m) nd fused qurtz (E6 Ohm.m). Becuse of this liner reltion between current density nd electric field there is lso liner reltion between current through device (I) nd voltge cross (V) the terminls of device. To double the voltge, you double the chrge which doubles the E, which double the J, which doubles the I. Note tht conductivity nd resistivity re mterils prmeters while resistnce (R) nd conductnce (G) re device prmeters: V IR I GV [] e cll this Ohm s lw lso it is not rel lw much more rule of thumb. e derived reltion between current density nd electric field using microscopic model, clculting first the ccelertion from the pplied electric field, turning tht into n ccelertion of the electrons. e furthermore ssumed tht the drift velocity of the electrons is proportionl to the men pth between two collisions of n electron. The men pth is considered to be mterils property nd depend on the concentrtion of sctter centre (collisions with grin boundries, defects, impurities, or lttice vibrtions (phonons)). As the therml velocity is much lrger thn the drift velocity, the men time between two collisions is independent of the of the drift velocity of the electrons, nd the verge drift velocity is liner dependent on the ccelertion nd thus the pplied electric field. e found: nfq J nfqvdrift E [3] mv therml here n is the number of toms per unit of volume, f the number of electrons per tom, q the chrge of n electron, m the effective mss of n electron, nd the men pth of the electrons. So electrons will hve lrge therml velocity nd then much smller drift velocity. Applying n electric field to mteril will ccelerte the electrons until the electrons collide with n imperfection in the mteril (think in terms of crystl structure) which will reset the drift velocity bck to zero. At the collision the kinetic energy of the electrons in converted into het, i.e. vibrtion of the toms: P VI I R [4]
2 e studied the I-V reltion of devices with different geometry (Problem 7.), i.e. the conventionl rectngulr resistor, the coxil cble, nd sphericl device. To find the I-V reltion we ssume chrge on the electrodes, use Guss lw to determine the electric field in the device, nd then use eqution [] nd the reltion between J nd I to find the current going through the device by integrting over n pproprite cross sectionl re: I J d E d [5] Note tht this pproch is very similr to the method used to determine the cpcitnce of device in chpter (exmple.). Exmple 7.3 shows tht for rectngulr devices with plnr electrodes the electric field is homogeneous through the device. e lso looked to other geometries, i.e. P7.39 nd e defined electromotive force of circuit loop: f dl [6] here f is the totl force per unit chrge on positive chrge consisting of two prts, i.e. n electrosttic prt E nd pushing prt of the source f s. The ltter contins lso ny emf generted by vrying mgnetic flux cught by the loop. In source E=-fs, so the voltge cross the terminls of bttery is: V b E dl b f dl s [7] Although btteries nd voltge nd current sources crete locl non zero emf, the emf cused by flux chnge through the loop is distributed cross the loop. e lerned tht chnging mgnetic field through the loop, or moving the loop wy from the mgnetic field, or chnging the size of the loop ll result in n induced emf in the loop. e defined flux rule: d [8] dt And from this flux rule we derived Frdy s lw in integrl nd differentil form: E dl B E t B d t [9] e studied vrious inductors, i.e. solenoid nd torroid, nd noticed tht chnging current through system of wires result in chnging mgnetic field nd thus results in n electric field, i.e. induced emf. According to Ampere s lw B is proportionl to current. The proportionl constnt for device ws
3 clled the inductnce of the device. e defined two different inductnces, i.e. the self-inductnce L, to describe the reltion between the current through the device nd the flux generted by the device: LI [0] And the mutul inductnce, M, to describe the flux coupling between two devices: [] M I e derived n expression to clculte the flux through device from the mgnetic vector potentil A: B d A dl Note tht we hve seen this reltion before in chpter 5, i.e. exmple 5.. This reltion cn be used to derive n expression for the mutul inductnce of device: M o 4 dl dl r s This is clled the Neumnn formul, lthough not so useful for most inductnce problems, it shows tht M =M, nd furthermore it shows tht M only depends on the geometry of the device. e studied how to determine the self inductnce or mutul inductnce of set of wires, i.e. first determine B from the current using Ampere s lw nd then integrting the B cross suitble crosssectionl re to find the flux (exmple 7.0). e lso derived n eqution for the induced emf of n inductor: di L [4] dt hich with the proper sign convention (i.e. voltge drop in the direction of the current similr to tht of resistor) will result in the following I-V reltion for n inductor: di V L [5] dt An expression for the energy stored in n inductor ws derived from eqution [4]. e hve seen this expression in PHYS45: [] [3] LI I [6] Substituting the inductor version of [] in [6] nd rewriting gives nd lterntive expression for the mgnetosttic energy:
4 o V B d S A B d [7] If the integrl is tken over the whole volume the surfce integrl becomes zero s A nd B re going to zero t lrge distnce from the source currents. So this results in two different mgnetosttic energy expressions: o ll spce ll spce B d A Jd [8] For the nd eqution one often only integrtes over the re where the current density is unequl to zero. e found tht term needs to be dded to Ampere s lw for Mxwell s equtions to be consistent with the continuity eqution. The totl set of Mxwell s eqution in differentil form become: E o B 0 B E t E B oj oo t [9] Those equtions together with the force lw: F q E v B [0] summrize the entire theoreticl content of clssicl electrodynmics nd even imply the continuity eqution i.e. J t [] Mxwell s equtions in mtter re expressed in terms of the uxiliry fields, D= o E+P nd H=/ o B-M, i.e.
5 D B 0 B E t H J D t [] The nd term in the Ampere Mxwell eqution is clled the displcement current density, this is lso bound current density but should not be confused with the bound current densities originting from the mgnetiztion we discussed in chpter 5, i.e Jb M K M nˆ b Note tht we rrived t these Mxwell equtions fter substituting the chrges nd current densities in eqution [9] by: [3] J J B P J bound J polrizti on J P M t [3] [3b] Note tht if the polriztion of cube is vrying s function of the time the bound surfce chrge t the top nd bottom of the cube increses. Associted with this bound surfce chrge is current. e refer to this current s the polriztion current nd it origintes from the electric dipole moment to be time dependent. Such time dependent polriztion will result in net trnsport of chrge from the bottom to the top surfce of the cube (ssuming polriztion is in the up direction in the cube). For liner mterils the constitutive equtions re: P oee M H m [4] Or D E H B [5] The boundry conditions for the em field were summrized in section 7.3.6:
6 n K H H E E B B D D ˆ 0 0 [6]
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